Boundary Value Problems Of Elastostatics Research Articles

Solution of a mixed nonaxisymmetric problem of the theory of elasticity for anisotropic bodies of revolution

The paper developed a technique for solving mixed nonaxisymmetric problems of the theory of elasticity for bounded bodies of revolution made of a transversely isotropic material under the action of surface forces specified according to a cyclic law. The technique involves the development of the energy method of boundary states, which is based on the concepts of spaces of internal and boundary states, conjugated by isomorphism, which makes it possible to establish a one-to-one correspondence between the elements of these spaces. The internal state includes the components of the tensor of stresses, deformations, and the displacement vector. The boundary state includes efforts and displacements at the boundary of the body. The isomorphism of the state spaces is proved, which allows finding the internal state to be reduced to the study of the boundary state isomorphic to it. The basis is formed on the basis of the general solution of the boundary value problem of elastostatics for a transversely isotropic body of revolution. Orthogonalization of state spaces is carried out, where the internal energy of elastic deformation is used as scalar products in the space of internal states; in the space of boundary states, the work of external forces is used. Finally, finding the desired state is reduced to solving an infinite system of algebraic equations for the Fourier coefficients. The solution of the problem with mixed boundary conditions for a circular in plan cylinder of transversely isotropic coarse dark gray siltstone with anisotropy axis coinciding with the geometric axis of symmetry is presented. The solution is analytical and the characteristics of the stress-strain state have a polynomial form. Explicit and indirect signs of convergence of problem solutions and graphical visualization of the results are presented.

Boundary value problems in elastostatics with singular data

We consider themain boundary value problems of linear elastostatics with nonregular data. We prove existence and uniqueness results for bounded and exterior domains of ℝ3 of class Ck (k ⩾ 2). In the case of isotropic body, we prove the results for domains of class C1,α (α ∈ (0, 1]) and of class C1 in the case of the displacement problem.

Explicit Solution of Elastostatic Boundary Value Problems for the Elastic Circle with Voids

We solve the static two-dimensional boundary value problems for an elastic porous circle with voids. Special representations of a general solution of a system of differential equations are constructed via elementary functions which make it possible to reduce the initial system of equations to equations of simple structure and facilitate the solution of the initial problems. Solutions are written explicitly in the form of absolutely and uniformly converging series. The question pertaining to the uniqueness of regular solutions of the considered problems is investigated.

Design optimization under uncertainties of a mesoscale implant in biological tissues using a probabilistic learning algorithm

This paper deals with the optimal design of a titanium mesoscale implant in a cortical bone for which the apparent elasticity tensor is modeled by a non-Gaussian random field at mesoscale, which has been experimentally identified. The external applied forces are also random. The design parameters are geometrical dimensions related to the geometry of the implant. The stochastic elastostatic boundary value problem is discretized by the finite element method. The objective function and the constraints are related to normal , shear, and von Mises stresses inside the cortical bone. The constrained nonconvex optimization problem in presence of uncertainties is solved by using a probabilistic learning algorithm that allows for considerably reducing the numerical cost with respect to the classical approaches.

Explicit solutions of the problems of elastostatics for an elastic circle with double porosity

In this paper the Aifantis linear theory of consolidation with double porosity is considered. The boundary value problems (BVPs) of elastostatics for an elastic circle are formulated and the uniqueness theorems for regular (classical) solutions are proved. The explicit solutions of these BVPs are constructed by means of absolutely and uniformly convergent series.

Uniqueness of the nonlinear elastic dielectric affine boundary value problem on the whole space and on cone-like regions

Conservation laws derived from the energy–momentum tensor are employed to establish under suitable sufficient conditions uniqueness in affine boundary value problems for the homogeneous nonlinear elastic dielectric on the whole space and on certain cone-like regions. In particular, the electric enthalpy is assumed to be strictly quasi-convex for the whole space, and strictly rank-one convex for cone-like regions. Asymptotic behaviour is also stipulated. Uniqueness results for corresponding affine boundary value problems of homogeneous nonlinear elastostatics are a special case of those derived here.

Introduction to Boundary Value Problems of Nonlinear Elastostatics

This paper provides a careful and accessible exposition of an $L^p$ approach to boundary value problems of nonlinear elastostatics in the case where solutions of the linearized problem correspond faithfully to those of the nonlinear problem, that is, in the case where there is no bifurcation. We prove that if the linearized problem has unique solutions, then so does the nonlinear one, nearby. This is done by using the linear $L^p$ theory and the inverse mapping theorem. The main theorem can be applied to the Saint Venant-Kirchhoff elastic material and the Hencky-Nadai elastoplastic material in a unified theory. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations.

Stress concentration factor due to a circular hole in functionally graded panels under uniaxial tension

The effect of the material property inhomogeneity on the stress concentration factor (SCF) due to a circular hole in functionally graded panels is numerically investigated. The multiple isoparametric finite element formulation is used to simulate the elastostatic boundary value problem. A parametric study is performed by varying the functional form and the direction of the material property gradation. The material property inhomogeneity is characterized by the intrinsic inhomogeneity length scale, modulus ratio and the power-law index. The results from our parametric study showed that the SCF is reduced when Young's modulus progressively increased away from the hole. The angular position of the maximum tensile stress on the surface of the hole remains unaffected by the material property inhomogeneity. The SCF is seen to be most influenced by the power-law index, followed by the variation of the inhomogeneity length scale. The SCF is least affected by the modulus ratio.

On the uniqueness of solutions for the Dirichlet boundary value problem of linear elastostatics in the circular domain

The uniqueness and mathematical stability of the Dirichlet boundary value problem of linear elastostatics is studied. The problem is posed as a set of partial differential equations in terms of displacements and Dirichlet-type of boundary conditions (displacements) for arbitrary bounded domains. Then for the circular interior domain the closed form analytical solution is obtained, using an extended version of the method of separation of variables. This method with corresponding complete solution allows for the derivation of a necessary and sufficient condition for uniqueness. The results are compared with existing energy and uniqueness criteria. A parametric study of the elastic characteristics is performed to investigate the behaviour of the displacement field and the strain energy distribution, and to examine the mathematical stability of the solution. It is found that the solution for the circular element with hourglass-like boundary conditions will be unique for all v ≠ 0.5, 0.75, 1.0 and will be mathematically stable for all v ≠ 0.75. Locking of the circular element occurs for v = 0.75 as the energy tends to infinity

Some Aspects of the BEM Research in China

In this paper, a short historical review of the BEM research in China is given first. The investigation on the BEM in China was started in 1978 at the Tsinghua University, based on the pioneering work published by Frank Rizzo [F. J. Rizzo, An integral equation approach to boundary value problems of classical elastostatics, Quart. Appl. Math., 25, 83-95 (1967)] and others. During the last 25 years, we have organized six national conferences on BEM, two international conferences and eight China-Japan/Japan-China symposia on BEM. The BEM investigations in the authors’ group at Tsinghua are briefly introduced next. Some representative investigations by other groups in Mainland China are also introduced. The related papers and other publications are listed in the References.

Application of the generalized perturbation-based stochastic boundary element method to the elastostatics

This paper is entirely devoted to the demonstration of a solution for some boundary value problems of isotropic linear elastostatics with random parameters using the boundary element method. The stochastic perturbation technique in its general nth-order Taylor series expansion version is used to express all the random parameters and the state functions of the problem. These expansions inserted in the classical deterministic equilibrium statement return up to the nth-order (both PDEs and matrix) equations. Contrary to the previous implementations of the stochastic perturbation technique, any order partial derivatives with respect to the random input are derived from the deterministic structural response function (SRF) at a given point. This function is approximated using polynomials by the least-squares method from the multiple solution of the initial deterministic problem solved for the expectations of random structural parameters. First two probabilistic moments have been computed symbolically here using the computational MAPLE environment, also as the polynomial expressions including perturbation parameter ε. It should be mentioned that such a generalized perturbation approach makes it possible to analyze all types of random variables (not only Gaussian) and to compute even higher probabilistic moments with a priori given accuracy. The entire methodology can be implemented after minor modifications to analyze nonlinear phenomena for both statics and dynamics of even heterogeneous domains.

Signorini’s method for live loads and second-order effects

In this paper we deal with Signorini’s method for live loads to derive approximate solutions of some pure traction boundary value problems in finite elastostatics. Then, the obtained solutions are used to determine the second-order constitutive constants of a homogeneous and isotropic elastic material.

A Fast Multipole Boundary Integral Equation Method for Periodic Boundary Value Problems in Three-Dimensional Elastostatics and its Application to Homogenization

A Fast Multipole Boundary Integral Equation Method for Periodic Boundary Value Problems in Three-Dimensional Elastostatics and its Application to Homogenization

Applications of a fast multipole Galerkin in boundary element method in linear elastostatics

Boundary element methods provide a powerful tool for solving boundary value problems of linear elastostatics, especially in complicated three–dimensional structures. In contrast to the standard Galerkin approach leading to dense stiffness matrices, in fast boundary element methods such as the fast multipole method the application of matrix–vector products can be realized with almost linear complexity. Since all boundary integral operators of linear elastostatics can be reduced to those of the Laplacian, the discretization of the corresponding single and double layer potentials of the Laplace operator has to be employed only. This technique results in a fast multipole method which is an efficient tool for the simulation of elastic stress fields in engineering and industrial applications.

The inverse deformation mapping in the finite element method

This paper aims at the exploitation of material forces to find an optimum mesh in the finite element method (FEM). The classical variational formulation provides the linear momentum equation in a Lagrangian description. A variational setting for the derivation of the canonical momentum equation in the Eulerian description is presented. The latter is based on an extremum principle for the total potential energy functional defined in terms of the inverse deformation function. This constitutes a theoretical framework which allows the formulation of the finite element method for the canonical momentum equation as well as the computation of the material forces arising from the discretization. Thus, apart from the finite element solution for the standard boundary value problem of elastostatics, a second one for the canonical momentum equation can be formulated and solved numerically. The former provides an optimum deformation by minimizing the standard total potential energy, namely solving the physical forces equilibrium equation. The latter provides an optimum discretization by minimizing the total potential energy in terms of the inverse deformation function, that is, solving the material force equilibrium equation. The latter provides an optimum discretization by minimizing the total potential energy in terms of the inverse deformation function, that is, solving the material forces equilibrium equation. The theoretical considerations are supported by providing a computational example.

Handbook of Green’s Functions and Matrices

7R2. Handbook of Green’s Functions and Matrices. - VD Seremet (State Agrarian Univ of Moldova, Rep of Moldova). WIT Press, Southampton, UK. Distributed in USA by Comput Mech, Billerica MA. 2003. 295 pp. CD-Rom included. ISBN 1-85312-933-X. $229.00.Reviewed by GC Gaunaurd (Code AMSRL-SE-RU, Army Res Lab, 2800 Powder Mill Rd, Adelphi MD 20783-1197).This handbook is intended for the specialist interested in the solution of Poisson’s equation and the Navier equations of elastostatics (here called the Lame´ equations) in a number of particular situations. The handbook comprises a book and a compact disc (CD). The book has two parts each with two chapters. Part I contains the theoretical part. Part II gives a list of ‘Green’s functions’ for Poisson’s equation and a list of ‘dyadic Green’s functions’ (here called Green’s matrices) for the Navier equations of elastostatics. (These are not to be confused with the Navier-Stokes equations of Fluid Mechanics, which are time-dependent and nonlinear, since they are expressed in Eulerian coordinate frames.) Part III is a CD that contains six appendices to the book, with numerous practical examples. Appendix I shows how to construct Green’s functions for Poisson’s equation, and Appendices III and V, the ones for the Navier equations. The other three appendices give examples of cases in which these Green’s functions appear in applied problems of mathematical physics. It is difficult to summarize this book in this brief review and do it justice. The author has done an excellent job. It covers material not well known in the West. As far as this reviewer knows, this book is unique in this respect. These Green’s functions are the basic tools needed to solve boundary value problems in elastostatics. Part I introduces the method of “incompressible influence elements.” This method leads to a new theory developed by the author for the construction of influence functions for bulk-dilatation and dyadic Green’s functions, whose components are the displacement components for boundary-value problems of elastostatics. Part II implements the theory presented in Part I and constructs the Green’s functions of some specific boundary-value problems (BVP). Here is a practical collection of hundreds of problems on Green’s functions for these equations. These problems and their answers are presented to the reader for use in practical cases or to practice his own mathematical skills. The derivations are not shown; only the answers are listed. As mentioned above, Part III is the CD. Here many typical 2D and 3D BVPs for Poisson’s and Navier equations are solved. Some explanations on how to construct the Green’s Function in question are given here. The book is a collection of Green’s functions for two elliptic second-order partial differential equations, and as such, is quite heavy on the math. There are no graphs, and there are also some shortcomings. For example, only the first basic problem of elasticity is addressed, only domains describable in Cartesian coordinate systems are presented, and there are many grammatical errors due to perhaps the fact that the author is not very familiar with the technical terminology in English. Handbook of Green’s Functions and Matrices is quite excellent, and in many respects, unique. Its intended audience is the advanced graduate student, and since there are no derivations, maybe it is intended for the research specialist, who already knows how to derive these results, but wishes to have quick access to them. In other words, it is like a Table of Integrals. Its steep price will undoubtedly restrict its circulation. It is certainly a reference of high value for institutional libraries. This reviewer believes that it would be advantageous if the author would some day extend this work to the elastodynamic (ie, time-dependent) cases governed by the various types of wave-equations and/or the Navier-equations of elastodynamics that govern elastic vibrations. These dynamic situations are more likely to be encountered in practical instances, rather than their static counterparts.

311 直接法による静弾性方程式の境界要素解 II

311 直接法による静弾性方程式の境界要素解 II

830 直接法による静弾性方程式の境界要素解

830 直接法による静弾性方程式の境界要素解

427 領域内変位観測を用いた静弾性境界値問題に対するチホノフの適切化法の適用と適切化パラメータの選定

427 領域内変位観測を用いた静弾性境界値問題に対するチホノフの適切化法の適用と適切化パラメータの選定

Boundary eigensolutions in elasticity. I. Theoretical development

The theory of fundamental boundary eigensolutions for elastostatic boundary value problems is developed. The underlying fundamental eigenproblem is formed by inserting the eigenparameter and a tensor weight function into the boundary condition, rather than into the governing differential equation as is often done for vibration problems. The resulting spectra are real and the eigenfunctions (eigendeformations) are mutually orthogonal on the boundary, thus providing a basis for solutions. The weight function permits effective treatment of non-smooth problems associated with cracks, notches and mixed boundary conditions. Several ideas related to the behavior of eigensolutions in the domain, integral equation methods, variational methods, convergence characteristics, flexibility and stiffness kernels, and solutions to problems with body forces are also introduced. Of particular note are the integral equation and variational formulations that lead to the development of new computational formulations for boundary element and finite element methods, respectively. An example with closed form and numerical results is included to illustrate some aspects of the theory.